Multi-material compressible flows are observed in several applications of practical relevance including combustion, cavitation and shock lithotripsy. To solve such flows numerically with improved efficiency and accuracy, the use of the Ghost Fluid Method (GFM) and its derivatives have been adopted widely. However, certain drawbacks of the ghost fluid approach such as overheating errors, and the appearance of unphysical oscillations of density and pressure are well known and have been documented in the literature. In this work, a cut-cell based GFM approach is proposed to accurately solve for the fluid state at the multi-material interface, while the location of the interface is tracked by a second-order level set scheme. The modified GFM was implemented in IMPACT, a multi-material, shock-physics code. The improvements to the GFM are validated through a wide range of simulations including multi-material flow problems, 1D shock problems, 2D shock-induced bubble collapse in water/air and the 2D Richtmyer-Meshkov instability. In addition, the efficiency of the numerical solver is improved with the implementation of a blocked-structured Adaptive Mesh Refinement (AMR) scheme. We find the AMR implementation enhances computational efficiency, while improving the mass conservation properties of the level set scheme.